Question

problem 1
a normal distribution has a mean of 50 and a standard deviation of 5. given that the area under the curve to the left of 45 is 0.1587 and the area to the left of 55 is 0.8413, find each area described below. explain your reasoning.

1. find the area under the curve to the right of 45.
2. find the area under the curve between 45 and 55.
3. what proportion of values fall within one standard deviation of the mean (between 45 and 55)? express as a percentage.

problem 2
the weight of a population of seals is approximately normally distributed with a mean of 80 kg and a standard deviation of 8 kg. using technology or a standard normal table, estimate the proportion (as a decimal and percentage) of seals in each category.

1. what proportion of seals weigh less than 88 kg?
2. what proportion of seals weigh between 72 kg and 96 kg?
3. what proportion of seals weigh more than 96 kg?

Explanation:

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Problem 1

1. Area right of 45

Step1: Total area = 1, subtract left area

$1 - 0.1587 = 0.8413$

2. Area between 45 and 55

Step1: Subtract left area of 45 from left area of 55

$0.8413 - 0.1587 = 0.6826$

3. Proportion within 1 standard deviation

Step1: Convert decimal to percentage

$0.6826 \times 100 = 68.26\%$

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Problem 2

First, calculate z-scores with formula $z = \frac{x - \mu}{\sigma}$, where $\mu=80$, $\sigma=8$

1. Proportion less than 88 kg

Step1: Calculate z-score for 88

$z = \frac{88 - 80}{8} = 1$

Step2: Find area left of z=1

From standard normal table, area = 0.8413; percentage = $0.8413 \times 100 = 84.13\%$

2. Proportion between 72 kg and 96 kg

Step1: Calculate z-scores for 72 and 96

$z_1 = \frac{72 - 80}{8} = -1$, $z_2 = \frac{96 - 80}{8} = 2$

Step2: Subtract left areas of z=-1 from z=2

Area left of z=-1=0.1587, area left of z=2=0.9772; $0.9772 - 0.1587 = 0.8185$

Step3: Convert to percentage

$0.8185 \times 100 = 81.85\%$

3. Proportion more than 96 kg

Step1: Use area left of z=2, subtract from 1

$1 - 0.9772 = 0.0228$

Step2: Convert to percentage

$0.0228 \times 100 = 2.28\%$

Answer:

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Problem 1

1. $0.8413$
2. $0.6826$
3. $68.26\%$

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Problem 2

1. Decimal: $0.8413$, Percentage: $84.13\%$
2. Decimal: $0.8185$, Percentage: $81.85\%$
3. Decimal: $0.0228$, Percentage: $2.28\%$